Parameters estimate of Riemannian Gaussian distribution in the manifold of covariance matrices

Abstract

The study of Pm, the manifold of m × m symmetric positive definite matrices, has recently become widely popular in many engineering applications, like radar signal processing, mechanics, computer vision, image processing, and medical imaging. A large body of literature is devoted to the barycentre of a set of points in Pm and the concept of barycentre has become essential to many applications and procedures, for instance classification of SPD matrices. However this concept is often used alone in order to define and characterize a set of points. Less attention is paid to the characterization of the shape of samples in the manifold, or to the definition of a probabilistic model, to represent the statistical variability of data in Pm. Here we consider Gaussian distributions and mixtures of Gaussian distributions on Pm. In particular we deal with parameter estimation of such distributions. This problem, while it is simple in the manifold P2, becomes harder for higher dimensions, since there are some quantities involved whose analytic expression is difficult to derive. In this paper we introduce a smooth estimate of these quantities using convex cubic splines, and we show that in this case the parameters estimate is coherent with theoretical results. We also present some simulations and a real EEG data analysis.